Theory & Simulation

Core and affiliate faculty of the 2DCC can provide expertise in a wide range of physio-chemical aspects of two-dimensional materials growth, characterization, and properties. These efforts can be predictive – preceding associated experimental efforts, or interpretive – assisting in the understanding of existing experimental efforts. As a 2DCC User, you can request specific theory/simulation capabilities in your user proposal, or later as your project proceeds and new opportunities emerge, perhaps through interactions with 2DCC staff. We do have certain overall resource limitations, so access to a specific capability is dependent on availability of personnel and associated resources to support their work.

Reactive Force Fields · Computational Fluid Dynamics · Phase-Field Methods · First-Principles Methods · Defect Electronic Structure ·  Defects and Optical Response · Data-Centric Approaches

1. The ReaxFF Reactive Force Fields: Force Field Development and MD Simulation

The Reactive Force Field known as ReaxFFis a framework for empirical atomistic dynamical simulations of growing materials, including chemical reactionsthat may take place in the gas phase or on the surface, for systems up to ~1,000,000 atoms and over timescales not accessible to first-principles techniques. The 2DCC makes extensive use of reactive force fields and continues to extend their application to more complex and realistic synthetic environments, with the codes hosted at the Penn State Material Computation Center. ReaxFF is supported in LAMMPSand can be used for molecular dynamics (NVE, NVT, NPT, μVT ensembles), energy minimization, Monte Carlo, hybrid Monte Carlo (i.e. Force Biased Monte Carlo3) and hyperdynamics/metadynamics acceleration methods such as Parallel Replica4,5, Parallel Tempering6, Collective Variable Hyperdynamicsand Bond Boost Dynamics8,9.

Specific implementations of ReaxFF are generally developed for each materials systems of interest, guided by and validated against a combination of first-principles and experimental data. Materials systems and substrates of potential interest to 2DCC users for which implementations currently exist include: 

  • MoS210,11
  • WSe(unpublished) 
  • Graphene + defects, including plasma treatment11,12
  • Plasma-assisted defect generation in graphene13
  • hBN (on metal surfaces)14,15
  • MXenes, specifically TiC16-19with a Mo extension under development
  • Phosphorene20
  • Silicene21-23
  • Sulfur (solid and molecular)24
  • Selenium (solid and molecular) (unpublished)
  • MoO(solid and molecular)25
  • Molecular precursors used in chemical vapor deposition: Mo(CO)6, W(CO)6, H2S, and H2Se
  • Hydrocarbon molecules (H/C/O)
  • Gallium and Indium metals, including interaction with Si.
  • GaN

Reactive force fields are currently under development for MoSe2, WS2and associated alloy systems.  Certain substrates can also be simulated, including

  • SiO226
  • Al2O327
  • SiC28-29
  • Graphene, including defects and interactions with sulfur and MoS211

These substrates can be used for physicalinteractions with all 2D-materials. Reactiveinteractions between a substrate and growing 2D layer involve additional force field development. This level of theory is currently under development for  MoS2/SiO2, followed by extension to other Mo/W/Se/S phases and additional substrates.

Possible applications of ReaxFF to your synthesis efforts could include direct simulation of the complex kinetics involved at the substrate surface to guide the choice of substrate or growth conditions for MBE or CVD, chemical kinetics in the gas phase prior to arrival of species at the substrate to better inform choices of precursors for MOCVD, or simulation of interfacial reactivity in a heterostructure with any growth method. The computational efficiency of these methods enable the generation of very large datasets of complex kinetic phenomena that are then amenable to further analysis using machine learning methods. Non-reactive phenomena such as elastic response can also be accessed. 2DCC staff also have experience with registry-dependent interlayer potentials (of the Kolmogorov-Crespi form30) that can handle complex relaxations in e.g. twisted bilayers. Some of these applications may be clear at the advent of your user project, and others may only arise midstream through your project.

2. Computational Fluid Dynamics

MOCVD is a complex highly coupled phenomenon, with many critical steps in synthesis potentially happening in the gas phase and further complex kinetics once the substrate gets involved. Physical and chemical phenomena occur across a wide range of length scales and time scales, from the macroscale flow of gases in the reactor chamber to the atomic-scale nucleation and growth of the resulting 2D material. 2DCC is developing a multi-scale computational approach which combines density functional calculations, reactive molecular dynamics, and full-scale computational fluid dynamics to better understand these processes30. The team is developing a detailed gas-phase chemical kinetic model to describe major chemical reaction pathways from precursors (for example, W(CO)and H2Se used to synthesize WSe2) to the thermodynamically stable end product, applying density functional theory to key gas-phase reactions to train and validate a ReaxFF force field that is then deployed to map out reaction pathways and energetics. Other kinetic parameters, plus thermal and transport properties of all species involved, are estimated from established kinetic/collisional theories.

This chemical kinetic model then informs computational fluid dynamics (CFD) simulations of gas-phase physio-chemical processes in the MOCVD chamber under experimental operating conditions, using a reacting flow solver31that captures gas-phase chemical reactions, gas-phase mass and heat transport, variable-density fluid dynamics, the transition from momentum-driven to buoyancy-driven flow, substrate rotation, and conjugate heat transfer. At this level of simulation, the CFD model can provide important insights into which reactive species actually impinge on the growth surface: these can vary dramatically from those injected into the chamber due to gas-phase reactions that proceed inhomogeneously as the reactants proceed through different physical conditions in their flow across the substrate. Parametric CFD simulations can determine the sensitivity of desired growth processes to operating conditions such as chamber pressure, substrate temperature, reactant injection speed, and reactant injection concentration. The team is currently focusing on gas-phase behavior for W(CO)and H2Se precursors in the growth of WSe2, and will extend to other systems as time and resources allow. Once chemical species contact the substrate, further complex chemistry ensues, with additional kinetic processes involved in transport of species along the substrate, at the growth edge, and on top of already-grown layers; these processes are currently being characterized for eventual integration with CFD models. Our intention is for high-fidelity theory and simulation to provide key insights to expand the scope of what is possible in terms of composition, doping, and material quality, with machine learning methods guiding the optimization to obtain desired growth outcomes.

Similar methods can be applied to powder vapor transport growth of transition metal dichalcogenides, where the precise location of the substrate can have pronounced effects on the morphology of the resulting films, changing from isolated 2D islands, to a multigrain monolayer film, and vertical 2D materials.The 2DCC team can couple reactor-scale heat and mass transport equations with mesoscale phase-field equations for growth morphology considering the variation of edge energies with precursor concentration within the growth chamber to predict the spatial distributions of 2D islands32-35.

3. Mesoscale phase-field models

Phase-field methods, currently under development, can describe island morphology evolution during growth of 2D materials taking into account the interplay of anisotropic edge energies, substrate/edge diffusivities, deposition rates and defect-island interactions. In these simulations, atomistic calculations provide information on edge energies, defect-island interactions and computational fluid dynamics simulations provide information on the initial precursor concentrations and deposition rates under different CVD growth conditions. Kinetic parameters such as diffusion barriers, adsorption energies and edge energies extracted from first-principles simulationscan also be fed into other larger-scale models such as kinetic Monte Carlo.

4. First-principles modeling of growth

Density functional methods can provide the most accurate atomistic-level treatment of the relative energetics of different geometries that may be present during growth, and can also provide guidance on the relative thermodynamic stability of competing phases. Specific key geometries, defined through a combination of insights gleaned from experiment and empirical modeling, can be analyzed in great depth to identify viable candidates for microscopic mechanisms of key steps during growth. For example, metal atoms coordinated to vacancy defects in the substrate have been identified through first-principles modeling to potentially play a key role in lifting the near degeneracy between certain 180-degree misaligned polar domains on incommensurate polar substrates36. Experimental observation often inspire new calculations. For example, fascinating modulations in the composition within 2D layers (e.g. striping of Mo and W atoms within growing transition metal dichalcogenide monolayers) can motivate computational investigations that reveal how fluctuations in the availability of chalcogenide at the growth edge can modulate the favorability to incorporate different metals, even when the bulk thermodynamics barely distinguishes the two species.

5. Data-Centric Methods

The 2DCC is partnering with MaterialsWeb.org to provide 2DCC users with detailed information on a comprehensive database of over 700 two-dimensional materials identified through high-throughput identification of low exfoliation energies from known bulk compounds37. The database currently provides the exfoliation energy, electronic band structure, and Pourbaix diagram for each material. The full database is freely available, and development is proceeding on extensions to incorporate defect energetics and other information on materials of specific interest to 2DCC users, integrated into the 2DCC user web access portal. Grand canonical genetic algorithm methods can further predict the composition and atomic structure of new 2D materials beyond those resident in the current database38

image5.pngExisting high-high-throughput databases are generally built around equilibrium or near-equilibrium thermodynamic properties of materials, while empirical methods handle more complex growth kinetics. The 2DCC team is pur suing high-throughput screening of optimal 2D material / substrate combinations by correlating key characteristics of kinetic growth models at the atomistic level39,40 with substrate/precursor interactions that can be derived from high-throughput models, to provide insights on the choice of substrate when growing a target material. Data-centric methods are also infused through the other theory/modeling capabilities described on this page – please refer to the other sections to learn more.

 

6. Defects and Electronic Structure

image14.pngElectronic states associated with structural defects in 2D materials often dominate their optical response, limit the performance possible in electronic and optoelectronic applications, and impede the discernment of ideal characteristics and exotic quantum correlated phenomena at low temperature. Computations of such defects is particularly complex when they carry a non-zero charge. The formation energy must account for corrections due to electrostatic interactions with periodic images and compensating background charges in the vacuum which are introduced in supercell calculations; i.e. we must restores the appropriate electrostatic boundary conditions for charged 2D materials41. Properly corrected calculations can provide defect formation energies in 2D materials such as MoSand phosphorene, from which one can determine the charge transition levels, i.e., the positions of defect within the band gap. Detailed microscopic insights regarding specific defects can also be obtained from density functional calculations. For example, Jahn-Teller distortions can stabilize the charged states of chalcogenide vacancies in certain 2D materials by breaking the symmetry otherwise present between certain dorbitals. Furthermore, chemisorption of ligands to the surface of 2D layers can profoundly affect the ability to dope carriers into the layers, and the physisorpimage9.pngtion of various species is likewise strongly coupled to the charge state of the monolayer. Additional examples abound where carefully constructed first-principles methods provide insights into key aspects of experimental systems. The 2DCC team plans to develop an automated workflow to expedite the setup and post-processing of first-principles calculations of charged point defects in 2D materials and use it to create a new database of defect properties in 2D and layered bulk materials. The development of this workflow will enable the rapid and robust evaluation of defect structures, formation energies, and electronic properties for different types of defects in a wide range of 2D materials.

7. Defects and Excited States

image10.pngThe 2DCC team can also model the excited state properties of pristine or defective 2D systems at the GW (including quasiparticle effects) and BSE levels (including excitonic effects). Using recently developed techniques42 that drastically accelerate convergence for GW calculations, we can extend these computationally demanding tasks to defective structures. In addition, time-dependent DFT methods parametrized from GW calculations43 can further accelerate excited-state calculations to enable the high-throughput screening of defect properties. Experimentally measured properties accessible to (potentially high-throughput) computation include optical absorption spectroscopy, electron energy loss spectroscopy, resonant or non-resonant Raman intensities, and second-order nonlinear susceptibility (second harmonic generation). This work is proceeding with partners at Cal State Northridge.

8. Infrastructure Partners

The 2DCC is supported by partnership with the Materials Computation Center (MCC) and the Institute for CyberScience – Advanced Cyberinfrastructure (ICS-ACI) at Penn State.

The MCC supports simulations of materials across multiple length and time scales, with technical support of advanced simulation software. The 2DCC is implementing, in partnership with the MCC, an integrated suite of software tools, hardware resources, and computational expertise to support direct simulation of key processes during materials growth, to suggest routes to overcome experimental obstacles, to assist in the interpretation of in situ characterization and post-synthesis measurement of samples, and also to predict new synthesis targets. 2DCC users will have access to MCC capabilities in computational methods including first-principles techniques at the density functional level (Quantum Espresso, VASP, etc.) and powerful empirical methods that can model complex reaction pathways over long time scales and length scales at both atomistic (ReaxFF) and phase-field levels.

One layer below MCC in the high-performance computation stack is the ICS-ACI, which acquires, hosts, and supports high-performance computational hardware and software. 2DCC computational resources will reside and be maintained by ICS-ACI. The ICS-ACI central facility serves the entire campus and is located at Penn State’s University Park Campus in the Penn State Data Center. The total computational resource of the ICS-ACI currently include:

  • Compute Resources - 23,000 basic-, standard- and high-memory nodes (128 GB, 256 GB, and 1 TB) with dual 10- or 12-core Xeon E5-2680 processors.
  • Storage - Over 2.5 PB of network-attached storage (NAS) to support users’ Home, Work, and Group storage needs and 2.5 PB of general parallel file system (GPFS) for Scratch storage in addition to 4 PB of tape storage for backup purposes. 
  • Network - High-performance Network Fabric built on Brocade VCS fabric technology. The architecture employs a research-centric software stack and customized environments allow researchers to deploy software from pre-compiled and tested software catalogs. The software stack supports both commercial and Opensource software.
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